#72303721

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10995
commutator	C₂×#41	C₂×C₄²
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#123590
prime radical	C₂×#3449	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

4941 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,9,15)(3,16,21,46)(4,22,28,58)(5,14,8,12)(6,30,7,13)(10,32,35,18)(11,36,43,60)(17,50,55,37)(19,48,20,49)(23,33,53,62)(24,38,29,44)(25,41,27,40)(26,39,31,42)(34,56,45,51)(47,59,57,61)(52,63,54,64);(1,3)(2,10)(4,23,28,53)(5,18,8,32)(6,20,7,19)(9,21)(11,37,43,50)(12,16,14,46)(13,45,30,34)(15,35)(17,26)(22,57,58,47)(24,51)(25,59)(27,61)(29,56)(31,55)(33,40)(36,54,60,52)(38,48)(39,63)(41,62)(42,64)(44,49);(1,4,24,60,9,28,29,36)(2,11,38,58,15,43,44,22)(3,17,51,59,21,55,56,61)(5,27,7,26,8,25,6,31)(10,33,48,63,35,62,49,64)(12,42,30,40,14,39,13,41)(16,47,34,50,46,57,45,37)(18,52,19,53,32,54,20,23)

Polycyclic presentation:

2^10|a^2=j,b^2=h,c^2=eg,d^2=j,e^2=j,b^a=bd,c^a=cdf,c^b=ce,d^a=dj,d^b=di,d^c=di,e^a=ei,e^b=eij,e^c=ej,e^d=ej,f^b=fij,f^c=fij,f^e=fj,g^a=gi,g^b=gij,g^c=gj,g^d=gj,g^f=gj,h^a=hij,h^c=hi,h^d=hj,h^e=hj,h^f=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303732, #72312305, #72312744, …
maximal normal subgroup of	#72303732, #72312305, #72312744, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.5*F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);